Manipulating topology in tailored artificial graphene nanoribbons

Topological phases of matter give rise to exotic physics that can be leveraged for next generation quantum computation and spintronic devices. Thus, the search for topological phases and the quantum states that they exhibit have become the subject of a massive research effort in condensed matter physics. Topologically protected states have been produced in a variety of systems, including artificial lattices, graphene nanoribbons (GNRs) and bismuth bilayers. Despite these advances, the real-time manipulation of individual topological states and their relative coupling, a necessary feature for the realization of topological qubits, remains elusive. Guided by first-principles calculations, we spatially manipulate robust, zero-dimensional topological states by altering the topological invariants of quasi-one-dimensional artificial graphene nanostructures. This is achieved by positioning carbon monoxide molecules on a copper surface to confine its surface state electrons into artificial atoms positioned to emulate the low-energy electronic structure of graphene derivatives. Ultimately, we demonstrate control over the coupling between adjacent topological states that are finely engineered and simulate complex Hamiltonians. Our atomic synthesis gives access to an infinite range of nanoribbon geometries, including those beyond the current reach of synthetic chemistry, and thus provides an ideal platform for the design and study of novel topological and quantum states of matter.

In this work, we investigate the emergence and interplay between SPT states in topologically non-trivial artificial GNRs through the real-time manipulation of their geometries and consequently their Z2 invariants. Additionally, we control the coupling between adjacent states by adjusting their separation and show how it can be used to simulate complex Hamiltonians and quantum states.
Tailored artificial graphene nanostructures are constructed by first using first-principles calculations to identify a GNR which exhibits a non-trivial Z2 invariant. The Z2 invariant is determined from the origin independent part of the Zak phase as 2 = graphene in an analogous way to how GNRs differ from pristine graphene, namely that they are laterally confined. A unique "stitching" was developed for confinement at the edges of our artificial structures.
The bulk-boundary correspondence dictates that boundary modes arise at the interface between a TI and trivial insulator, namely where there is a change in the Z2 invariant 26 . We demonstrate this effect at the edge of a topologically non-trivial N=9 armchair GNR (Z2=1) and vacuum (Z2=0) (Fig. 2a). The configuration of the atomic sites at the termination uniquely determines the nanoribbon unit cell, the origin of which determines the Z2 invariant in onedimension 19 . In Fig 2b, we demonstrate the switch between a topologically non-trivial and trivial nanoribbon by constructing an identical 9GNR but removing three artificial atoms from one zigzag edge and adding them to the other side (Fig. 2b). The origin independent part of the Zak phase of the modified unit cell in Fig 2b ( 2 ′ ) is related to the original unit cell ( 2 ) as 2 where is the number of new (or old) lattice sites added (or removed) 23 (Supplementary Information). The topological class of the nanoribbon is changed from Z2=1 to Z2=0 since we move an odd number of sites effectively homogenizing its Z2 invariant with the vacuum. To visualize the emergence of boundary modes along the non-trivial interface, we acquire simultaneous STM topographies and normalized differential conductance maps at -55meV of the topologically non-trivial (Fig. 2c) and trivial (Fig. 2d) nanoribbon. The SPT states that decorate the zig-zag edge of the non-trivial nanoribbon vanish when the structure becomes topologically trivial, corroborated by the local density of states (LDOS) maps calculated by tight-binding (TB) (bottom right panels). The LDOS at the zig-zag edge atoms are further investigated by acquiring dI/dV point spectroscopy (Fig. 2e). The spectrum on the non-trivial edge shows a resonance around -55meV that is not present on the trivial structure suggesting that it emerges from the spatial modulation of its topological character in agreement with the TB LDOS.
To distinguish topology from other mechanisms that could produce edge states we replace the vacuum with a 7GNR assembly. The resulting 7GNR/9GNR heterostructure contains artificial atoms along the interface with the same coordination as those in the bulk. The termination of the 7GNR segment, and hence the choice of unit cell, depends on whether the heterostructure is symmetric (Fig. 3a) or asymmetric (Fig. 3b). The topological class of the symmetric and asymmetric unit cells is Z2=0 and Z2=1, respectively (Supplementary Information). These results are in agreement with Ref 19 and illustrate the relationship between band topology and nanoribbon unit-cell. To visualize the interface states, STM topographies and normalized differential conductance maps were acquired at -30meV over the artificial atoms of the non-trivial (Fig 3c) and trivial (Fig. 3d) interface. In agreement with the calculated LDOS maps (bottom panels), the conductance of the two nanoribbons is similar everywhere except for the interface. The dichotomy between the two interfaces is further exemplified in the magnified conductance maps of the nontrivial ( Fig. e and trivial (Fig. 3f) interfaces. Line profiles extracted from these maps over the first two rows of interfacial artificial atoms show the states present over the interface where the Z2 invariant changes that vanish when both sides of the interface are topologically equivalent (Fig.   3g). Furthermore, the dI/dV point spectroscopy acquired on the artificial atom at the center of the interface in Figure 3e shows a resonance around -30meV that is absent in the spectrum acquired over the analogous atom in Fig. 3f (Fig. 3f), in good agreement with the TB model.
To examine the range of coupling achievable between adjacent topological states and their robustness, we reduce the length of the middle segment LBC separating B and C while increasing the length of the end segment LCD separating C and D, thereby maintaining a constant total length ( Fig. 4b). By moving state C towards a fixed state B and acquiring a line profile over the central resonance of the latter (Fig. 4c), the increased coupling between the two interface states leads to a decrease in the intensity of the peaks at both interfaces. As intensity is proportional to the square of the local components of the eigenstates, such a decrease in intensity demonstrates increased delocalization and the tunability of the wavefunction of the eigenstates resulting from the SPTs.
Notably, the coupling between the interface state and end state results in a qualitatively distinct effect with measurable differences between the state B and C only occurring when LCD < 3 unit cells (Fig. 4d). These trends are captured by the TB model (right panel in Fig. 4b and dashed lines in Fig. 4c, d).
The interplay between the four states and the effect of the different segment lengths can be   Supplementary Information is available for this paper.
Correspondence and requests for materials should be addressed to nguisinger@anl.gov   Supplementary Tables 1 to 5 Supplementary Reference List

Materials and Methods
The STM measurements were performed using an ultra-high vacuum (1. Spectroscopy of the symmetry protected topological (SPT) states were measured using a standard lock-in technique with a modulation voltage of 20mV and a modulation frequency of 1.125kHz.

Muffin Tin Calculations
The arrangement of CO molecules results in a change in potential landscape of the free electron like surface states of Cu. We follow the approach of Kempkes et. al. [20]  The parameters for the tight binding model are then obtained by fitting (discussed below) the band dispersion with that computed from Muffin Tin calculations

Tight Binding Calculations
The lattice formed by electronic states confined by CO molecules is modeled under tight binding (TB) approximation. The artificial atoms or the electronic confinement (anti-lattice of the CO arrangement) is considered as a single electron orbital in TB with their corresponding positions.
We include the interaction of the sites with the nearest ( ) and next nearest ( ) neighbor. All The LDOS from TB is calculated by applying a Lorentzian function (equation (3)) with a broadening of b = 30 meV to account for the scattering. Our value of the parameter b is obtained empirically and based on the width of the experimentally measured spectra.

Parameterization of tight binding parameters
The tight binding band dispersion obtained from the Hamiltonian of the 2D periodic unit cell shown in Supplementary Figure 1 is fitted to the muffin tin dispersion (Supplementary Figure 2).
The fitting parameters are , , and s. The values obtained using a non-linear least squares fit through Levenberg-Marquardt algorithm as implemented in the curve_fit() function of scipy package [28] is shown in Supplementary Table 1. These parameters will be used for the rest of the discussion.

Topological classification
The different artificial GNR geometries studied in experiments are classified as topologically nontrivial or topologically trivial based on the Z2 invariant [17] defined as As noted by Lin et. al [17], the above definition of the Z2 invariant is independent of the real space origin but still depends on the unit cell definition. The definition of the unit cell is uniquely determined by the termination of the finite nanoribbon [11,12,16,17]. Since the GNRs are 1D periodic, the above integral can be numerically computed as the sum of Wannier Charge Centers  Table 2 shows the different unit cell geometry considered in this work and their respective Z2 invariant. Table 2, moving three atoms in the unit cell (effectively changing the termination of the semi-infinite nanoribbon), changes the topological class. Following the conventions used in Ref. [17], the total Zak phase for any 1D periodic structure can be written as

As shown in Supplementary
While 1, depends on the expectation value of , 2, is independent of the real space origin. 2, can be interpreted as the intercell Zak phase which depends on the choice of the unit cell but independent of the choice of the real space origin [16,24]. Additionally, we note that the Z2 invariant is related to 2, as 2 = ∑ 2, 2. In general, when M new (or old) lattice sites are added (or removed) to the unit cell definition of the nanoribbon, the total intercell Zak phase of the modified unit cell ( 2 ′ = ∑ 2, ′ ) is related to the original unit cell as 2 Hence, when M is odd, we have a change in the topological class and when M is even the topological class is preserved (see Supplementary Table 2).

Effective Hamiltonian of topological states.
The Supplementary Table 3 lists the values of computed using the procedure described above. We note that decays exponentially with length and can be approximated as an exponential fit = − as shown in Supplementary Figure 8.
While is found to be finite at = 1 and decays exponentially, we find that and are decoupled ( =0) for all the lengths (see Supplementary Table 4). This phenomenon can be Additionally, by symmetry, we have = = 0.
Next, we calculate the onsite energies using equation (11) and the corresponding values are listed in Supplementary Table 4. This explains the energy shift observed in the TB modeling for the "3-5-1" and "3-4-2" configuration, as the 9GNR segment that state B is always in contact with has a length 3-unit cells, while state C is in contact with a relatively short 9GNR segment. We conjecture that the reduced interaction between the 9GNR HOMO and the SPT state due to the larger energy difference between these states in the short segment (state C) leads to further stabilization of the SPT in state C that can be understood in second-order perturbation theory on the energy. This conclusion is 24 also in agreement with the experimental observation of a more localized wavefunction on state C for the 3-5-1 configuration (Fig. 4 in the main text).